Magento – How to sort product collection on stock status

I want to sort product collection on category page so that the in stock products shall appear before out of stock products. All the out of stock products will be displayed after the in stock products. I have already rewritten the Mage_Product_Block_List block. After rewriting I have tried the following:

protected function _getProductCollection()
{
    if (is_null($this->_productCollection)) {
        $layer = $this->getLayer();
        /* @var $layer Mage_Catalog_Model_Layer */
        if ($this->getShowRootCategory()) {
            $this->setCategoryId(Mage::app()->getStore()->getRootCategoryId());
        }

        // if this is a product view page
        if (Mage::registry('product')) {
            // get collection of categories this product is associated with
            $categories = Mage::registry('product')->getCategoryCollection()
                ->setPage(1, 1)
                ->load();
            // if the product is associated with any category
            if ($categories->count()) {
                // show products from this category
                $this->setCategoryId(current($categories->getIterator()));
            }
        }

        $origCategory = null;
        if ($this->getCategoryId()) {
            $category = Mage::getModel('catalog/category')->load($this->getCategoryId());
            if ($category->getId()) {
                $origCategory = $layer->getCurrentCategory();
                $layer->setCurrentCategory($category);
                $this->addModelTags($category);
            }
        }
        **$this->_productCollection = $layer->getProductCollection()->addAttributeToSort('stock_status', desc);**

        $this->prepareSortableFieldsByCategory($layer->getCurrentCategory());

        if ($origCategory) {
            $layer->setCurrentCategory($origCategory);
        }
    }

    return $this->_productCollection;
}